Intended Models of ZFC: in response to http://www.askphilosophers.org/question/1935

[kate]

I read the response by Velleman to this question (http://
www.askphilosophers.org/question/1935) and I'd be interested if
somebody could please clarify a couple of points about the
interpretation of set membership.

I want to know whether any countable model of ZFC could interpret set
membership in the "standard" way, and also get a clarification of what
counts as standard (transitive, and...what else?). You can get a
countable transitive model for ZFC, which also proves that some sets
are nondenumerable in a Skolem's-paradoxy way. What else could make
models "intended" than transitivity? If we interpret membership as
transitive, even that isn't going to be completely "intended" because
its just restricted to range over the domain of the model in question.
So is this where/why unrestricted quantification is appealed to?

I read Velleman's response to a similar question, and now I have even
more questions! So, suppose we have a countable model of ZFC: when you
say its domain *is* the natural numbers, do you mean we literally swap
each element for an element of \omega? Because, my understanding of
the natural numbers is that they're sets, so when Velleman writes "In
this model, the "is an element of" symbol would be interpreted as some
relation on the natural numbers", I'm a bit puzzled, because set
membership itself is a relation on the naturals.

Anybody got any ideas about the first question?
Thankyou,
Kate